// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009-2011, 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_MATRIX_FUNCTION_H
#define EIGEN_MATRIX_FUNCTION_H

#include "StemFunction.h"

namespace Eigen {

namespace internal {

/** \brief Maximum distance allowed between eigenvalues to be considered "close". */
static const float matrix_function_separation = 0.1f;

/** \ingroup MatrixFunctions_Module
 * \class MatrixFunctionAtomic
 * \brief Helper class for computing matrix functions of atomic matrices.
 *
 * Here, an atomic matrix is a triangular matrix whose diagonal entries are close to each other.
 */
template<typename MatrixType>
class MatrixFunctionAtomic
{
  public:
	typedef typename MatrixType::Scalar Scalar;
	typedef typename stem_function<Scalar>::type StemFunction;

	/** \brief Constructor
	 * \param[in]  f  matrix function to compute.
	 */
	MatrixFunctionAtomic(StemFunction f)
		: m_f(f)
	{
	}

	/** \brief Compute matrix function of atomic matrix
	 * \param[in]  A  argument of matrix function, should be upper triangular and atomic
	 * \returns  f(A), the matrix function evaluated at the given matrix
	 */
	MatrixType compute(const MatrixType& A);

  private:
	StemFunction* m_f;
};

template<typename MatrixType>
typename NumTraits<typename MatrixType::Scalar>::Real
matrix_function_compute_mu(const MatrixType& A)
{
	typedef typename plain_col_type<MatrixType>::type VectorType;
	Index rows = A.rows();
	const MatrixType N = MatrixType::Identity(rows, rows) - A;
	VectorType e = VectorType::Ones(rows);
	N.template triangularView<Upper>().solveInPlace(e);
	return e.cwiseAbs().maxCoeff();
}

template<typename MatrixType>
MatrixType
MatrixFunctionAtomic<MatrixType>::compute(const MatrixType& A)
{
	// TODO: Use that A is upper triangular
	typedef typename NumTraits<Scalar>::Real RealScalar;
	Index rows = A.rows();
	Scalar avgEival = A.trace() / Scalar(RealScalar(rows));
	MatrixType Ashifted = A - avgEival * MatrixType::Identity(rows, rows);
	RealScalar mu = matrix_function_compute_mu(Ashifted);
	MatrixType F = m_f(avgEival, 0) * MatrixType::Identity(rows, rows);
	MatrixType P = Ashifted;
	MatrixType Fincr;
	for (Index s = 1; double(s) < 1.1 * double(rows) + 10.0; s++) { // upper limit is fairly arbitrary
		Fincr = m_f(avgEival, static_cast<int>(s)) * P;
		F += Fincr;
		P = Scalar(RealScalar(1) / RealScalar(s + 1)) * P * Ashifted;

		// test whether Taylor series converged
		const RealScalar F_norm = F.cwiseAbs().rowwise().sum().maxCoeff();
		const RealScalar Fincr_norm = Fincr.cwiseAbs().rowwise().sum().maxCoeff();
		if (Fincr_norm < NumTraits<Scalar>::epsilon() * F_norm) {
			RealScalar delta = 0;
			RealScalar rfactorial = 1;
			for (Index r = 0; r < rows; r++) {
				RealScalar mx = 0;
				for (Index i = 0; i < rows; i++)
					mx = (std::max)(mx, std::abs(m_f(Ashifted(i, i) + avgEival, static_cast<int>(s + r))));
				if (r != 0)
					rfactorial *= RealScalar(r);
				delta = (std::max)(delta, mx / rfactorial);
			}
			const RealScalar P_norm = P.cwiseAbs().rowwise().sum().maxCoeff();
			if (mu * delta * P_norm < NumTraits<Scalar>::epsilon() * F_norm) // series converged
				break;
		}
	}
	return F;
}

/** \brief Find cluster in \p clusters containing some value
 * \param[in] key Value to find
 * \returns Iterator to cluster containing \p key, or \c clusters.end() if no cluster in \p m_clusters
 * contains \p key.
 */
template<typename Index, typename ListOfClusters>
typename ListOfClusters::iterator
matrix_function_find_cluster(Index key, ListOfClusters& clusters)
{
	typename std::list<Index>::iterator j;
	for (typename ListOfClusters::iterator i = clusters.begin(); i != clusters.end(); ++i) {
		j = std::find(i->begin(), i->end(), key);
		if (j != i->end())
			return i;
	}
	return clusters.end();
}

/** \brief Partition eigenvalues in clusters of ei'vals close to each other
 *
 * \param[in]  eivals    Eigenvalues
 * \param[out] clusters  Resulting partition of eigenvalues
 *
 * The partition satisfies the following two properties:
 * # Any eigenvalue in a certain cluster is at most matrix_function_separation() away from another eigenvalue
 *   in the same cluster.
 * # The distance between two eigenvalues in different clusters is more than matrix_function_separation().
 * The implementation follows Algorithm 4.1 in the paper of Davies and Higham.
 */
template<typename EivalsType, typename Cluster>
void
matrix_function_partition_eigenvalues(const EivalsType& eivals, std::list<Cluster>& clusters)
{
	typedef typename EivalsType::RealScalar RealScalar;
	for (Index i = 0; i < eivals.rows(); ++i) {
		// Find cluster containing i-th ei'val, adding a new cluster if necessary
		typename std::list<Cluster>::iterator qi = matrix_function_find_cluster(i, clusters);
		if (qi == clusters.end()) {
			Cluster l;
			l.push_back(i);
			clusters.push_back(l);
			qi = clusters.end();
			--qi;
		}

		// Look for other element to add to the set
		for (Index j = i + 1; j < eivals.rows(); ++j) {
			if (abs(eivals(j) - eivals(i)) <= RealScalar(matrix_function_separation) &&
				std::find(qi->begin(), qi->end(), j) == qi->end()) {
				typename std::list<Cluster>::iterator qj = matrix_function_find_cluster(j, clusters);
				if (qj == clusters.end()) {
					qi->push_back(j);
				} else {
					qi->insert(qi->end(), qj->begin(), qj->end());
					clusters.erase(qj);
				}
			}
		}
	}
}

/** \brief Compute size of each cluster given a partitioning */
template<typename ListOfClusters, typename Index>
void
matrix_function_compute_cluster_size(const ListOfClusters& clusters, Matrix<Index, Dynamic, 1>& clusterSize)
{
	const Index numClusters = static_cast<Index>(clusters.size());
	clusterSize.setZero(numClusters);
	Index clusterIndex = 0;
	for (typename ListOfClusters::const_iterator cluster = clusters.begin(); cluster != clusters.end(); ++cluster) {
		clusterSize[clusterIndex] = cluster->size();
		++clusterIndex;
	}
}

/** \brief Compute start of each block using clusterSize */
template<typename VectorType>
void
matrix_function_compute_block_start(const VectorType& clusterSize, VectorType& blockStart)
{
	blockStart.resize(clusterSize.rows());
	blockStart(0) = 0;
	for (Index i = 1; i < clusterSize.rows(); i++) {
		blockStart(i) = blockStart(i - 1) + clusterSize(i - 1);
	}
}

/** \brief Compute mapping of eigenvalue indices to cluster indices */
template<typename EivalsType, typename ListOfClusters, typename VectorType>
void
matrix_function_compute_map(const EivalsType& eivals, const ListOfClusters& clusters, VectorType& eivalToCluster)
{
	eivalToCluster.resize(eivals.rows());
	Index clusterIndex = 0;
	for (typename ListOfClusters::const_iterator cluster = clusters.begin(); cluster != clusters.end(); ++cluster) {
		for (Index i = 0; i < eivals.rows(); ++i) {
			if (std::find(cluster->begin(), cluster->end(), i) != cluster->end()) {
				eivalToCluster[i] = clusterIndex;
			}
		}
		++clusterIndex;
	}
}

/** \brief Compute permutation which groups ei'vals in same cluster together */
template<typename DynVectorType, typename VectorType>
void
matrix_function_compute_permutation(const DynVectorType& blockStart,
									const DynVectorType& eivalToCluster,
									VectorType& permutation)
{
	DynVectorType indexNextEntry = blockStart;
	permutation.resize(eivalToCluster.rows());
	for (Index i = 0; i < eivalToCluster.rows(); i++) {
		Index cluster = eivalToCluster[i];
		permutation[i] = indexNextEntry[cluster];
		++indexNextEntry[cluster];
	}
}

/** \brief Permute Schur decomposition in U and T according to permutation */
template<typename VectorType, typename MatrixType>
void
matrix_function_permute_schur(VectorType& permutation, MatrixType& U, MatrixType& T)
{
	for (Index i = 0; i < permutation.rows() - 1; i++) {
		Index j;
		for (j = i; j < permutation.rows(); j++) {
			if (permutation(j) == i)
				break;
		}
		eigen_assert(permutation(j) == i);
		for (Index k = j - 1; k >= i; k--) {
			JacobiRotation<typename MatrixType::Scalar> rotation;
			rotation.makeGivens(T(k, k + 1), T(k + 1, k + 1) - T(k, k));
			T.applyOnTheLeft(k, k + 1, rotation.adjoint());
			T.applyOnTheRight(k, k + 1, rotation);
			U.applyOnTheRight(k, k + 1, rotation);
			std::swap(permutation.coeffRef(k), permutation.coeffRef(k + 1));
		}
	}
}

/** \brief Compute block diagonal part of matrix function.
 *
 * This routine computes the matrix function applied to the block diagonal part of \p T (which should be
 * upper triangular), with the blocking given by \p blockStart and \p clusterSize. The matrix function of
 * each diagonal block is computed by \p atomic. The off-diagonal parts of \p fT are set to zero.
 */
template<typename MatrixType, typename AtomicType, typename VectorType>
void
matrix_function_compute_block_atomic(const MatrixType& T,
									 AtomicType& atomic,
									 const VectorType& blockStart,
									 const VectorType& clusterSize,
									 MatrixType& fT)
{
	fT.setZero(T.rows(), T.cols());
	for (Index i = 0; i < clusterSize.rows(); ++i) {
		fT.block(blockStart(i), blockStart(i), clusterSize(i), clusterSize(i)) =
			atomic.compute(T.block(blockStart(i), blockStart(i), clusterSize(i), clusterSize(i)));
	}
}

/** \brief Solve a triangular Sylvester equation AX + XB = C
 *
 * \param[in]  A  the matrix A; should be square and upper triangular
 * \param[in]  B  the matrix B; should be square and upper triangular
 * \param[in]  C  the matrix C; should have correct size.
 *
 * \returns the solution X.
 *
 * If A is m-by-m and B is n-by-n, then both C and X are m-by-n.  The (i,j)-th component of the Sylvester
 * equation is
 * \f[
 *     \sum_{k=i}^m A_{ik} X_{kj} + \sum_{k=1}^j X_{ik} B_{kj} = C_{ij}.
 * \f]
 * This can be re-arranged to yield:
 * \f[
 *     X_{ij} = \frac{1}{A_{ii} + B_{jj}} \Bigl( C_{ij}
 *     - \sum_{k=i+1}^m A_{ik} X_{kj} - \sum_{k=1}^{j-1} X_{ik} B_{kj} \Bigr).
 * \f]
 * It is assumed that A and B are such that the numerator is never zero (otherwise the Sylvester equation
 * does not have a unique solution). In that case, these equations can be evaluated in the order
 * \f$ i=m,\ldots,1 \f$ and \f$ j=1,\ldots,n \f$.
 */
template<typename MatrixType>
MatrixType
matrix_function_solve_triangular_sylvester(const MatrixType& A, const MatrixType& B, const MatrixType& C)
{
	eigen_assert(A.rows() == A.cols());
	eigen_assert(A.isUpperTriangular());
	eigen_assert(B.rows() == B.cols());
	eigen_assert(B.isUpperTriangular());
	eigen_assert(C.rows() == A.rows());
	eigen_assert(C.cols() == B.rows());

	typedef typename MatrixType::Scalar Scalar;

	Index m = A.rows();
	Index n = B.rows();
	MatrixType X(m, n);

	for (Index i = m - 1; i >= 0; --i) {
		for (Index j = 0; j < n; ++j) {

			// Compute AX = \sum_{k=i+1}^m A_{ik} X_{kj}
			Scalar AX;
			if (i == m - 1) {
				AX = 0;
			} else {
				Matrix<Scalar, 1, 1> AXmatrix = A.row(i).tail(m - 1 - i) * X.col(j).tail(m - 1 - i);
				AX = AXmatrix(0, 0);
			}

			// Compute XB = \sum_{k=1}^{j-1} X_{ik} B_{kj}
			Scalar XB;
			if (j == 0) {
				XB = 0;
			} else {
				Matrix<Scalar, 1, 1> XBmatrix = X.row(i).head(j) * B.col(j).head(j);
				XB = XBmatrix(0, 0);
			}

			X(i, j) = (C(i, j) - AX - XB) / (A(i, i) + B(j, j));
		}
	}
	return X;
}

/** \brief Compute part of matrix function above block diagonal.
 *
 * This routine completes the computation of \p fT, denoting a matrix function applied to the triangular
 * matrix \p T. It assumes that the block diagonal part of \p fT has already been computed. The part below
 * the diagonal is zero, because \p T is upper triangular.
 */
template<typename MatrixType, typename VectorType>
void
matrix_function_compute_above_diagonal(const MatrixType& T,
									   const VectorType& blockStart,
									   const VectorType& clusterSize,
									   MatrixType& fT)
{
	typedef internal::traits<MatrixType> Traits;
	typedef typename MatrixType::Scalar Scalar;
	static const int Options = MatrixType::Options;
	typedef Matrix<Scalar, Dynamic, Dynamic, Options, Traits::RowsAtCompileTime, Traits::ColsAtCompileTime>
		DynMatrixType;

	for (Index k = 1; k < clusterSize.rows(); k++) {
		for (Index i = 0; i < clusterSize.rows() - k; i++) {
			// compute (i, i+k) block
			DynMatrixType A = T.block(blockStart(i), blockStart(i), clusterSize(i), clusterSize(i));
			DynMatrixType B = -T.block(blockStart(i + k), blockStart(i + k), clusterSize(i + k), clusterSize(i + k));
			DynMatrixType C = fT.block(blockStart(i), blockStart(i), clusterSize(i), clusterSize(i)) *
							  T.block(blockStart(i), blockStart(i + k), clusterSize(i), clusterSize(i + k));
			C -= T.block(blockStart(i), blockStart(i + k), clusterSize(i), clusterSize(i + k)) *
				 fT.block(blockStart(i + k), blockStart(i + k), clusterSize(i + k), clusterSize(i + k));
			for (Index m = i + 1; m < i + k; m++) {
				C += fT.block(blockStart(i), blockStart(m), clusterSize(i), clusterSize(m)) *
					 T.block(blockStart(m), blockStart(i + k), clusterSize(m), clusterSize(i + k));
				C -= T.block(blockStart(i), blockStart(m), clusterSize(i), clusterSize(m)) *
					 fT.block(blockStart(m), blockStart(i + k), clusterSize(m), clusterSize(i + k));
			}
			fT.block(blockStart(i), blockStart(i + k), clusterSize(i), clusterSize(i + k)) =
				matrix_function_solve_triangular_sylvester(A, B, C);
		}
	}
}

/** \ingroup MatrixFunctions_Module
 * \brief Class for computing matrix functions.
 * \tparam  MatrixType  type of the argument of the matrix function,
 *                      expected to be an instantiation of the Matrix class template.
 * \tparam  AtomicType  type for computing matrix function of atomic blocks.
 * \tparam  IsComplex   used internally to select correct specialization.
 *
 * This class implements the Schur-Parlett algorithm for computing matrix functions. The spectrum of the
 * matrix is divided in clustered of eigenvalues that lies close together. This class delegates the
 * computation of the matrix function on every block corresponding to these clusters to an object of type
 * \p AtomicType and uses these results to compute the matrix function of the whole matrix. The class
 * \p AtomicType should have a \p compute() member function for computing the matrix function of a block.
 *
 * \sa class MatrixFunctionAtomic, class MatrixLogarithmAtomic
 */
template<typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex>
struct matrix_function_compute
{
	/** \brief Compute the matrix function.
	 *
	 * \param[in]  A       argument of matrix function, should be a square matrix.
	 * \param[in]  atomic  class for computing matrix function of atomic blocks.
	 * \param[out] result  the function \p f applied to \p A, as
	 * specified in the constructor.
	 *
	 * See MatrixBase::matrixFunction() for details on how this computation
	 * is implemented.
	 */
	template<typename AtomicType, typename ResultType>
	static void run(const MatrixType& A, AtomicType& atomic, ResultType& result);
};

/** \internal \ingroup MatrixFunctions_Module
 * \brief Partial specialization of MatrixFunction for real matrices
 *
 * This converts the real matrix to a complex matrix, compute the matrix function of that matrix, and then
 * converts the result back to a real matrix.
 */
template<typename MatrixType>
struct matrix_function_compute<MatrixType, 0>
{
	template<typename MatA, typename AtomicType, typename ResultType>
	static void run(const MatA& A, AtomicType& atomic, ResultType& result)
	{
		typedef internal::traits<MatrixType> Traits;
		typedef typename Traits::Scalar Scalar;
		static const int Rows = Traits::RowsAtCompileTime, Cols = Traits::ColsAtCompileTime;
		static const int MaxRows = Traits::MaxRowsAtCompileTime, MaxCols = Traits::MaxColsAtCompileTime;

		typedef std::complex<Scalar> ComplexScalar;
		typedef Matrix<ComplexScalar, Rows, Cols, 0, MaxRows, MaxCols> ComplexMatrix;

		ComplexMatrix CA = A.template cast<ComplexScalar>();
		ComplexMatrix Cresult;
		matrix_function_compute<ComplexMatrix>::run(CA, atomic, Cresult);
		result = Cresult.real();
	}
};

/** \internal \ingroup MatrixFunctions_Module
 * \brief Partial specialization of MatrixFunction for complex matrices
 */
template<typename MatrixType>
struct matrix_function_compute<MatrixType, 1>
{
	template<typename MatA, typename AtomicType, typename ResultType>
	static void run(const MatA& A, AtomicType& atomic, ResultType& result)
	{
		typedef internal::traits<MatrixType> Traits;

		// compute Schur decomposition of A
		const ComplexSchur<MatrixType> schurOfA(A);
		eigen_assert(schurOfA.info() == Success);
		MatrixType T = schurOfA.matrixT();
		MatrixType U = schurOfA.matrixU();

		// partition eigenvalues into clusters of ei'vals "close" to each other
		std::list<std::list<Index>> clusters;
		matrix_function_partition_eigenvalues(T.diagonal(), clusters);

		// compute size of each cluster
		Matrix<Index, Dynamic, 1> clusterSize;
		matrix_function_compute_cluster_size(clusters, clusterSize);

		// blockStart[i] is row index at which block corresponding to i-th cluster starts
		Matrix<Index, Dynamic, 1> blockStart;
		matrix_function_compute_block_start(clusterSize, blockStart);

		// compute map so that eivalToCluster[i] = j means that i-th ei'val is in j-th cluster
		Matrix<Index, Dynamic, 1> eivalToCluster;
		matrix_function_compute_map(T.diagonal(), clusters, eivalToCluster);

		// compute permutation which groups ei'vals in same cluster together
		Matrix<Index, Traits::RowsAtCompileTime, 1> permutation;
		matrix_function_compute_permutation(blockStart, eivalToCluster, permutation);

		// permute Schur decomposition
		matrix_function_permute_schur(permutation, U, T);

		// compute result
		MatrixType fT; // matrix function applied to T
		matrix_function_compute_block_atomic(T, atomic, blockStart, clusterSize, fT);
		matrix_function_compute_above_diagonal(T, blockStart, clusterSize, fT);
		result = U * (fT.template triangularView<Upper>() * U.adjoint());
	}
};

} // end of namespace internal

/** \ingroup MatrixFunctions_Module
 *
 * \brief Proxy for the matrix function of some matrix (expression).
 *
 * \tparam Derived  Type of the argument to the matrix function.
 *
 * This class holds the argument to the matrix function until it is assigned or evaluated for some other
 * reason (so the argument should not be changed in the meantime). It is the return type of
 * matrixBase::matrixFunction() and related functions and most of the time this is the only way it is used.
 */
template<typename Derived>
class MatrixFunctionReturnValue : public ReturnByValue<MatrixFunctionReturnValue<Derived>>
{
  public:
	typedef typename Derived::Scalar Scalar;
	typedef typename internal::stem_function<Scalar>::type StemFunction;

  protected:
	typedef typename internal::ref_selector<Derived>::type DerivedNested;

  public:
	/** \brief Constructor.
	 *
	 * \param[in] A  %Matrix (expression) forming the argument of the matrix function.
	 * \param[in] f  Stem function for matrix function under consideration.
	 */
	MatrixFunctionReturnValue(const Derived& A, StemFunction f)
		: m_A(A)
		, m_f(f)
	{
	}

	/** \brief Compute the matrix function.
	 *
	 * \param[out] result \p f applied to \p A, where \p f and \p A are as in the constructor.
	 */
	template<typename ResultType>
	inline void evalTo(ResultType& result) const
	{
		typedef typename internal::nested_eval<Derived, 10>::type NestedEvalType;
		typedef typename internal::remove_all<NestedEvalType>::type NestedEvalTypeClean;
		typedef internal::traits<NestedEvalTypeClean> Traits;
		typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
		typedef Matrix<ComplexScalar, Dynamic, Dynamic, 0, Traits::RowsAtCompileTime, Traits::ColsAtCompileTime>
			DynMatrixType;

		typedef internal::MatrixFunctionAtomic<DynMatrixType> AtomicType;
		AtomicType atomic(m_f);

		internal::matrix_function_compute<typename NestedEvalTypeClean::PlainObject>::run(m_A, atomic, result);
	}

	Index rows() const { return m_A.rows(); }
	Index cols() const { return m_A.cols(); }

  private:
	const DerivedNested m_A;
	StemFunction* m_f;
};

namespace internal {
template<typename Derived>
struct traits<MatrixFunctionReturnValue<Derived>>
{
	typedef typename Derived::PlainObject ReturnType;
};
}

/********** MatrixBase methods **********/

template<typename Derived>
const MatrixFunctionReturnValue<Derived>
MatrixBase<Derived>::matrixFunction(
	typename internal::stem_function<typename internal::traits<Derived>::Scalar>::type f) const
{
	eigen_assert(rows() == cols());
	return MatrixFunctionReturnValue<Derived>(derived(), f);
}

template<typename Derived>
const MatrixFunctionReturnValue<Derived>
MatrixBase<Derived>::sin() const
{
	eigen_assert(rows() == cols());
	typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar;
	return MatrixFunctionReturnValue<Derived>(derived(), internal::stem_function_sin<ComplexScalar>);
}

template<typename Derived>
const MatrixFunctionReturnValue<Derived>
MatrixBase<Derived>::cos() const
{
	eigen_assert(rows() == cols());
	typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar;
	return MatrixFunctionReturnValue<Derived>(derived(), internal::stem_function_cos<ComplexScalar>);
}

template<typename Derived>
const MatrixFunctionReturnValue<Derived>
MatrixBase<Derived>::sinh() const
{
	eigen_assert(rows() == cols());
	typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar;
	return MatrixFunctionReturnValue<Derived>(derived(), internal::stem_function_sinh<ComplexScalar>);
}

template<typename Derived>
const MatrixFunctionReturnValue<Derived>
MatrixBase<Derived>::cosh() const
{
	eigen_assert(rows() == cols());
	typedef typename internal::stem_function<Scalar>::ComplexScalar ComplexScalar;
	return MatrixFunctionReturnValue<Derived>(derived(), internal::stem_function_cosh<ComplexScalar>);
}

} // end namespace Eigen

#endif // EIGEN_MATRIX_FUNCTION_H
